

A167783


Numbers that are repdigits with length > 2 in more than one base.


3



31, 63, 255, 273, 364, 511, 546, 728, 777, 931, 1023, 1365, 1464, 2730, 3280, 3549, 3783, 3906, 4095, 4557, 6560, 7566, 7812, 8191, 9114, 9331, 9841, 10507, 11349, 11718, 13671, 14043, 14763, 15132, 15624, 16383, 18291, 18662, 18915, 19608, 19682, 21845, 22351, 22698
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OFFSET

1,1


COMMENTS

Definition requires "length > 2" because all numbers n > 2 are trivially represented as "11" in base n1.
From Daniel Forgues, Nov 13 2009: (Start)
0 = 00 = 000 = 0000 = 00000 = 000000 = 0000000 = 00000000 = ... in any positional number representation (includes fixed base radix b > 1, mixed base radix with each b_i > 1, i >= 0, such as factorial and primorial based radix...)
The sequence definition should be read as:
Nonnegative integers that are repdigits with length > 2 in more than one fixed base radix b > 1.
Considering all fixed and mixed base radix would include many more nonnegative integers (but not the integers 1 to 6) which are repdigits with length > 2 in more than one radix. (End)
From Bernard Schott, Aug 08 2017: (Start)
In this sequence data, the first number which is repdigit, with length > 2, in more than two bases is the twelth Mersenne number 4095 with four Brazilian representations: M_12 = 4095 = 111111111111_2 = 333333_4 = 7777_8 = (15 15 15)_16.
The Mersenne number M_15 is the first number which is repdigit in exactly three bases with M_15 = 32767 = 111111111111111_2 = 77777_8 = (31 31 31)_32.
The Mersenne number M_24 is (certainly) the smallest integer which is repdigit in exactly five bases with M_24 = 16777215 = [R_24]_12 = 333333333333_4 = 77777777_8 = (15 15 15 15 15 15)_16 = (63 63 63 63)_64.
Only two numbers are repunits in more than one base: the Mersenne primes 31 and 8191 (Examples and A119598).
Some numbers are once repunit and once multiple of a Brazilian prime such that Mersenne number M_9 = 511 = 7 * 73 = 111111111_2 = 7 * 111_8 = 777_8.
Some numbers are once repunit and once multiple of a composite repunit such that Mersenne number M_6 = 63 = 3 * 21 = 111111_2 = 3 * 111_4 = 333_4.
Some numbers are repdigits in two different bases: 546 = 666_9 = 222_16. (End)


LINKS

Michel Marcus, Table of n, a(n) for n = 1..158
Wolfram Demonstrations Project, Mixed Radix Number Representations


EXAMPLE

31 is in the list because 31 = 11111_2 = 111_5;
8191 = 1111111111111_2 = 111_90;
10507 = {19 19 19}_23 = 111_102.


MATHEMATICA

Select[Range[550], Function[n, 1 < Count[Range[2, n  1], _?(And[Length@ DeleteCases[#, 0] == 1, Union[#][[2]] > 2] &@ DigitCount[n, #] &)]]] (* Michael De Vlieger, Aug 09 2017 *)


PROG

(PARI) isok(n)=my(nb = 0); for (b=2, n1, d = digits(n, b); if ((#d > 2) && (#Set(d) == 1), nb++); if (nb > 1, return (1)); ); return (0); \\ Michel Marcus, Aug 08 2017


CROSSREFS

Cf. A167782 (numbers that are repdigits with length > 2 in some base).
Cf. A010785 (repdigits (base 10)).
Cf. A053696 (numbers which are repunits in some base).
Cf. A158235 (numbers n whose square is a repdigit in some base < n).
Sequence in context: A042910 A042908 A042912 * A042914 A042916 A042918
Adjacent sequences: A167780 A167781 A167782 * A167784 A167785 A167786


KEYWORD

nonn,base,changed


AUTHOR

Andrew Weimholt, Nov 12 2009


EXTENSIONS

a(41)a(44) from Bernard Schott, Aug 08 2017


STATUS

approved



